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CSC 421 Algorithm Design Analysis Spring 2013

- Greedy algorithms
- greedy algorithms
- examples optimal change, job scheduling
- Prim's algorithm (minimal spanning tree)
- Dijkstra's algorithm (shortest path)
- Huffman codes (data compression)
- applicability

Greedy algorithms

- the greedy approach to problem solving involves

making a sequence of choices/actions, each of

which simply looks best at the moment - local view choose the locally optimal option
- hopefully, a sequence of locally optimal

solutions leads to a globally optimal solution

- example optimal change
- given a monetary amount, make change using the

fewest coins possible - amount 16 coins?
- amount 96 coins?

Example greedy change

- while the amount remaining is not 0
- select the largest coin that is ? the amount

remaining - add a coin of that type to the change
- subtract the value of that coin from the amount

remaining - e.g., 96 50 25 10 10 1
- will this greedy algorithm always yield the

optimal solution?

- for U.S. currency, the answer is YES
- for arbitrary coin sets, the answer is NO
- suppose the U.S. Treasury added a 12 coin
- GREEDY 16 12 1 1 1 1 (5 coins)
- OPTIMAL 16 10 5 1 (3 coins)

Example job scheduling

- suppose you have a collection of jobs to execute

and know their lengths - want to schedule the jobs so as to minimize

waiting time - Job 1 5 minutes Schedule 1-2-3 0 5 15

20 minutes waiting - Job 2 10 minutes Schedule 3-2-1 0 4 14 18

minutes waiting - Job 3 4 minutes Schedule 3-1-2 0 4 9

13 minutes waiting - GREEDY ALGORITHM do the shortest job first
- i.e., while there are still jobs to execute,

schedule the shortest remaining job

does the greedy approach guarantee the optimal

schedule? efficiency?

Application minimal spanning tree

- consider the problem of finding a minimal

spanning tree of a graph - a spanning tree of a graph G is a tree (no

cycles) made up of all the vertices and a subset

of the edges of G - a minimal spanning tree for a weighted graph G is

a spanning tree with minimal total weight - minimal spanning trees arise in many real-world

applications - e.g., wiring a network of computers connecting

rural houses with roads

spanning tree? minimal spanning tree?

example from http//compprog.wordpress.com/

Prim's algorithm

- to find a minimal spanning tree (MST)
- select any vertex as the root of the tree
- repeatedly, until all vertices have been added
- find the lowest weight edge with exactly one

vertex in the tree - select that edge and vertex and add to the tree

Prim's algorithm

- to find a minimal spanning tree (MST)
- select any vertex as the root of the tree
- repeatedly, until all vertices have been added
- find the lowest weight edge with exactly one

vertex in the tree - select that edge and vertex and add to the tree

minimal spanning tree? is it unique?

Correctness of Prim's algorithm

- Proof (by induction) Each subtree T1, T1, ,

TV in Prim's algorithm is contained in a MST.

Thus, TV is a MST. - BASE CASE T1 contains a single vertex, so is

contained in a MST. - ASSUME T1, , Ti-1 are contained in a MST.
- STEP Must show Ti is contained in a MST.
- Assume the opposite, that Ti is not contained in

a MST. - Let ei be the new edge (i.e., minimum weight edge

with exactly one vertex in Ti-1). - Since we assumed Ti is not part of any MST,

adding ei to a MST will yield a cycle. - That cycle must contain another edge with exactly

one vertex in Ti-1 . - Replacing that edge with ei yields a spanning

tree, and since ei had the minimal weight of any

edge with exactly one vertex in Ti-1, it is a

MST. - Thus, Ti is contained in a MST ? CONTRADICTION!

Efficiency of Prim's algorithm

- brute force (i.e., adjacency matrix)
- simple (conservative) analysis
- for each vertex, must select the least weight

edge ? O(V E) - more careful analysis
- note that the number of eligible edges is

shrinking as the tree grows - S (V deg(vi)) O(V2 E) O(V2)

- smarter implementation
- use a priority queue (min-heap) to store

vertices, along with minimal weight edge - to select each vertex remove from PQ ? V

O(log V) O(V log V) - to update each adjacent vertex after removal (at

most once per edge) - E O(log V) O(E log V)
- overall efficiency is O( (EV) log V )

Application shortest path

- consider the general problem of finding the

shortest path between two nodes in a graph - flight planning and word ladder are examples of

this problem - - in these cases, edges have uniform cost

(shortest path fewest edges) - if we allow non-uniform edges, want to find

lowest cost/shortest distance path

Redville ? Purpleville ?

example from http//www.algolist.com/Dijkstra's_al

gorithm

Modified BFS solution

- we could modify the BFS approach to take cost

into account - instead of adding each newly expanded path to the

end (i.e., queue), add in order of path cost

(i.e., priority queue) - Redville0
- Redville, Blueville5,
- Redville, Orangeville8,
- Redville, Greenville10
- Redville, Orangeville8,
- Redville, Blueville, Greenville8,
- Redville, Greenville10,
- Redville, Blueville, Purpleville12
- Redville, Blueville, Greenville8,
- Redville, Greenville10,
- Redville, Orangeville, Purpleville10,
- Redville, Blueville, Purpleville12

note as before, requires lots of memory to store

all the paths HOW MANY?

Dijkstra's algorithm

- alternatively, there is a straightforward greedy

algorithm for shortest path - Dijkstra's algorithm
- Begin with the start node. Set its value to 0 and

the value of all other nodes to infinity. Mark

all nodes as unvisited. - For each unvisited node that is adjacent to the

current node - If (value of current node value of edge) lt

(value of adjacent node), change the value of the

adjacent node to this value. - Otherwise leave the value as is.
- Set the current node to visited.
- If unvisited nodes remain, select the one with

smallest value and go to step 2. - If there are no unvisited nodes, then DONE.
- this algorithm is O(N2), requires only O(N)

additional storage

Dijkstra's algorithm example

- suppose want to find shortest path from Redville

to Purpleville

- Begin with the start node. Set its value to 0 and

the value of all other nodes to infinity. Mark

all nodes as unvisited

- For each unvisited node that is adjacent to the

current node - If (value of current node value of edge) lt

(value of adjacent node), change the value of the

adjacent node to this value. - Otherwise leave the value as is.
- Set the current node to visited.

Dijkstra's algorithm example cont.

- If unvisited nodes remain, select the one with

smallest value and go to step 2. - Blueville set Greenville to 8 and Purpleville to

12 mark as visited. - Greenville no unvisited neighbors mark as

visited.

- If unvisited nodes remain, select the one with

smallest value and go to step 2. - Orangeville set Purpleville to 10 mark as

visited. - If there are no unvisited nodes, then DONE.

With all nodes labeled, can easily construct the

shortest path HOW?

Correctness efficiency of Dijkstra's algorithm

- analysis of Dijkstra's algorithm is similar to

Prim's algorithm - can show that each greedy selection is safe,

leads to shortest path - brute force (i.e., adjacency matrix) approach
- for each vertex, need to select shortest edge ?

O(V E) - or, more carefully, S (V deg(vi)) O(V2

E) O(V2) - smarter implementation
- use a priority queue (min-heap) to store

vertices, along with minimal weight edge - to select each vertex remove from PQ ? V

O(log V) O(V log V) - to update each adjacent vertex after removal ?

E O(log V) O(E log V) - overall efficiency is O( (EV) log V )

Another application data compression

- in a multimedia world, document sizes continue to

increase - a 6 megapixel digital picture is 2-4 MB
- an MP3 song is 3-6 MB
- a full-length MPEG movie is 800 MB
- storing multimedia files can take up a lot of

disk space - perhaps more importantly, downloading multimedia

requires significant bandwidth

- it could be a lot worse!
- image/sound/video formats rely heavily on data

compression to limit file size e.g., if no

compression, 6 megapixels 3 bytes/pixel 18

MB - the JPEG format provides 101 to 201 compression

without visible loss

Audio, video, text compression

- audio video compression algorithms rely on

domain-specific tricks - lossless image formats (GIF, PNG) recognize

repeating patterns (e.g. a sequence of white

pixels) and store as a group - lossy image formats (JPG, XPM) round pixel values

and combine close values - video formats (MPEG, AVI) take advantage of the

fact that little changes from one frame to next,

so store initial frame and changes in subsequent

frames - audio formats (MP3, WAV) remove sound out of

hearing range, overlapping noises - what about text files?

- in the absence of domain-specific knowledge,

can't do better than a fixed-width code - e.g., ASCII code uses 8-bits for each character
- '0' 00110000 'A' 01000001 'a' 01100001
- '1' 00110001 'B' 01000010 'b' 01100010
- '2' 00110010 'C' 01000011 'c' 01100011
- . . .
- . . .
- . . .

Fixed- vs. variable-width codes

- suppose we had a document that contained only the

letters a-f - with a fixed-width code, would need 3 bits for

each character - a 000 d 011
- b 001 e 100
- c 010 f 101
- if the document contained 100 characters, 100 3

300 bits required

- however, suppose we knew the distribution of

letters in the document - a45, b13, c12, d16, e9, f5
- can customize a variable-width code, optimized

for that specific file - a 0 d 111
- b 101 e 1101
- c 100 f 1100
- requires only 451 133 123 163 94

54 224 bits

Huffman codes

- Huffman compression is a technique for

constructing an optimal variable-length code

for text - optimal in that it represents a specific file

using the fewest bits - (among all symbol-for-symbol codes)
- Huffman codes are also known as prefix codes
- no individual code is a prefix of any other code
- a 0 d 111
- b 101 e 1101
- c 100 f 1100
- this makes decompression unambiguous

1010111110001001101 - note since the code is specific to a particular

file, it must be stored along with the compressed

file in order to allow for eventual decompression

Huffman trees

- to construct a Huffman code for a specific file,

utilize a greedy algorithm to construct a Huffman

tree - process the file and count the frequency for each

letter in the file - create a single-node tree for each letter,

labeled with its frequency - repeatedly,
- pick the two trees with smallest root values
- combine these two trees into a single tree whose

root is labeled with the sum of the two subtree

frequencies - when only one tree remains, can extract the codes

from the Huffman tree by following edges from

root to each leaf (left edge 0, right edge 1)

Huffman tree construction (cont.)

the code corresponding to each letter can be read

by following the edges from the root left edge

0, right edge 1 a 0 d 111 b 101 e

1101 c 100 f 1100

Huffman code compression

- note that at each step, need to pick the two

trees with smallest root values - perfect application for a priority queue

(min-heap) - store each single-node tree in a priority queue

(PQ) O(N log N) - repeatedly, O(N) times
- remove the two min-value trees from the PQ O(log

N) - combine into a new tree with sum at root and

insert back into PQ O(log N) - total efficiency O(N log N)

- while designed for compressing text, it is

interesting to note that Huffman codes are used

in a variety of applications - the last step in the JPEG algorithm, after

image-specific techniques are applied, is to

compress the resulting file using a Huffman code - similarly, Huffman codes are used to compress

frames in MPEG (MP4)

Greed is good?

- IMPORTANT the greedy approach is not applicable

to all problems - but when applicable, it is very effective (no

planning or coordination necessary) - GREEDY approach for N-Queens start with first

row, find a valid position in current row, place

a queen in that position then move on to the next

row

since queen placements are not independent, local

choices do not necessarily lead to a global

solution GREEDY does not work need a more

holistic approach